Regularized reduced rank regression for mixed predictor and response variables

In this paper, we introduce the Generalized Mixed Regularized Reduced Rank Regression model (GMR⁴), an extension of the GMR³ model designed to improve performance in high-dimensional settings. GMR³ is a regression method for a mix of numeric, binary and ordinal response variables, while also allowing for mixed-type predictors through optimal scaling. GMR⁴ extends this approach by incorporating regularization techniques, such as Ridge, Lasso, Group Lasso, or any combination thereof, making the model suitable for datasets with a large number of predictors or collinearity among them. In addition, we propose a cross-validation procedure that enables the estimation of the rank $S$ and the penalty parameter $\lambda$ . Through a simulation study, we evaluate the performance of the model under different scenarios, varying the sample size, the number of non-informative predictors and response dimension. The results of the simulation study guide the choice of the penalty parameter $\lambda$ in the empirical application ISSP: Health and Healthcare I-II (2023), which includes mixed-type predictors and ordinal responses. In this application, the model results in a sparse and interpretable solution, with a limited set of influential predictors that provide insights into public attitudes towards health care.